are algebraic numbers. These ratios of Γ functions occur in the asymptotic enumeration of certain classes of restricted partitions, but I don't think this is relevant. Also, In the partition problems I'm interested in, it's natural to have $r_1 + \ldots + r_j = s_1 + \ldots + s_j$ but this isn't necessary. This seems to happen with some frequency. For example,a note of Albert Nijenhuis (arXiv:0907.1689) shows that $\Gamma(1/14) \Gamma(9/14) \Gamma(11/14) = 4\pi^{3/2}$; the techniques of the same paper show that $\Gamma(3/14) \Gamma(5/14) \Gamma(13/14) = 2\pi^{3/2}$, so the quotient is in fact 2! Similarly, we can get the identity

to the first two factors in the numerator and the last two in the denominator. In trying to prove other identities of this type, the duplication formula, its generalization to the "multiplication formula"

are the most obvious tools. So this seems to be a problem in combinatorial number theory; given an expression of the form in the first displayed equation, when can we use the multiplication and reflection formulas to reduce it to a rational power of some integer times a product of trig functions of rational multiples of π?

I am pretty sure that it is even open whether $\Gamma(1/5)$ is irrational, much less algebraic, but I cannot find a reference.
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Boris BukhDec 3 '09 at 1:00

That's a good point. The question I'm interested in, I suppose, is when we can actually write down a polynomial satisfied by a ratio of gamma functions. (In other words, assume &Gamma;(z) is transcendental for all rational, non-integer z, which is morally true.)
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Michael LugoDec 3 '09 at 1:10

-1 until the question is made precise. Is $\Gamma(1/173)\Gamma(4/61)$ morally irrational? Etc.
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Boris BukhDec 3 '09 at 10:38